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Note on the strong convergence of a weighted sum
Journal of Inequalities and Applications volume 2014, Article number: 179 (2014)
Abstract
In this paper, we consider an interesting weighted sums , , where is a sequence of independent and identically distributed random variables with , and the equivalence of the almost sure and complete convergence of is proved.
1 Introduction
Hsu and Robbins [1] introduced the concept of complete convergence as follows: a sequence converges completely to the constant c if
They also show that the sequence of arithmetic means of i.i.d. random variables converges completely to the expected value if the variance of the summands is finite. The converse was proved by Erdös [2, 3]. Furthermore, if completely, then the Borel-Cantelli lemma trivially implies that , almost sure convergence as . The converse statement is generally, or even ‘typically’, not true.
There are numerous publications in the literature studying the almost sure convergence and complete convergence for the weighted sums of a sequence of random variables. Gut [4] provided necessary and sufficient conditions for the complete convergence of the Cesáro means of i.i.d. random variables. Li et al. [5] obtained some results on complete convergence for weighted sums of independent random variables. Cuzick [6] proved a strong law for weighted sums of i.i.d. random variables. Miao and Xu [7] established a general result for the weighted sums of stationary sequence.
Throughout this paper, assume that is a sequence of independent and identically distributed random variables with . Chow and Lai [8] established the following result.
Theorem 1.1 [[8], Theorem 3]
Let be i.i.d. random variables such that . For , the following statements are equivalent:
-
1.
;
-
2.
a.e.;
-
3.
a.e. for some (or equivalently for every) nonvoid sequence of real numbers (, ) such that , where .
As an interesting particular case of Theorem 1.1, we consider the following weighted sums:
The aim of this paper is to prove the following.
Theorem 1.2 We have if and only if completely.
Remark 1.1 For the case , Wu [9] discussed the following weighted sums:
and proved the equivalence of the almost sure and complete convergence of the sequence . On the one hand, because of the limitation of α in Theorem 1.1, we here only discuss the case of . On the other hand, in order to prove the equivalence of the almost sure and complete convergence of for the case , we need the exponential integrability of X, but from Theorem 1.1, this does not hold.
Let ; then we have the following.
Corollary 1.1 If , then
2 Proofs of main results
Let denote the usual integer part of ‘ ⋅ ’ and assume that . The constant C in the proofs below depends only on the distribution of the underlying random variable X and may denote different quantities at different appearances.
2.1 Proof of Theorem 1.2
The proof of Theorem 1.2 can be derived from Lemma 2.2 and Lemma 2.3.
Lemma 2.1 The following estimate holds:
Proof Since decreases in j,
which yields the desired result. □
Lemma 2.2 If , then
completely and
completely.
Proof From Theorem 1.1, we know , which implies . Hence by the elementary inequality
we have
Notice that if and , then . Hence, by Lemma 2.1, for any ,
Similarly, we can obtain , then . The first statement now follows if we combine the two inequalities. The same technique yields the second statement. □
Lemma 2.3 If
then completely.
Proof For any we have
where , . Since
for , the random variables , are independent. Notice also that . So by the Borel-Cantelli lemma and the second statement of Lemma 2.2, we have , proving the lemma. □
2.2 Proof of Corollary 1.1
We easily see that
Now we apply the inequality
(cf. Chow and Teicher [10], p.111) by taking for . Since for all n sufficiently large,
Therefore,
Replacing with , the corollary follows.
References
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Erdös P: On a theorem of Hsu and Robbins. Ann. Math. Stat. 1949, 20: 286–291. 10.1214/aoms/1177730037
Erdös P: Remark on my paper ‘On a theorem of Hsu and Robbins’. Ann. Math. Stat. 1950, 21: 138. 10.1214/aoms/1177729897
Gut A: Complete convergence and Cesáro summation for i.i.d. random variables. Probab. Theory Relat. Fields 1993, 97: 169–178. 10.1007/BF01199318
Li DL, Rao MB, Jiang TF, Wang XC: Complete convergence and almost sure convergence of weighted sums of random variables. J. Theor. Probab. 1995, 8: 49–76. 10.1007/BF02213454
Cuzick J: A strong law for weighted sums of i.i.d. random variables. J. Theor. Probab. 1995, 8: 625–641. 10.1007/BF02218047
Miao Y, Xu SF: Almost sure convergence of weighted sum. Miskolc Math. Notes 2013, 14: 173–181.
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Wu WB: On the strong convergence of a weighted sum. Stat. Probab. Lett. 1999, 1: 19–22.
Chow YS, Teicher H: Probability Theory. 3rd edition. Springer, New York; 1978.
Acknowledgements
This work is supported by HASTIT (No. 2011HASTIT011), NSFC (No. 11001077), NCET (NCET-11-0945), and Plan For Scientific Innovation Talent of Henan Province (124100510014).
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Li, X., Miao, Y. & Zhang, X. Note on the strong convergence of a weighted sum. J Inequal Appl 2014, 179 (2014). https://doi.org/10.1186/1029-242X-2014-179
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DOI: https://doi.org/10.1186/1029-242X-2014-179